Kalimuthu Degenerate Triangle Calculus (K-DTC) as a UV Regulator Tool for Regge Quantum Gravity: Bridging Validated Degenerated Spherical Triangles with Planck-Scale Discreteness
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Abstract
Background: In validation articles published in Annals of Mathematics and Physics [Kalimuthu 2024, 2025], we showed that Girard’s formula Area = (A+B+C-π) R² remains valid under Type I (side→0) and Type II (angle→π) degeneration. The tri-rectangular triangle (Sum=3π/2, E=π/2, Area=1/8·4πR²) was identified as a fundamental brick.
Methods: We develop Kalimuthu Degenerate Triangle Calculus (K-DTC) as a computational regulator for Regge quantum gravity. The validated excess E_deg = lim(Sum_deg-π) is identified rigorously as integrated Gaussian curvature/holonomy defect Φ_h concentrated at the hinge, corresponding to U(1) holonomy. We define K_eff = Φ_h / A*_h.
Results: K-DTC provides a finite limit even when A*_h→0. Maximal excess E_max=2π gives area bound 4lp² and curvature bound K_max= (π/2)/lp². Numerical tests on S² and S³ show convergence O (N⁻¹) and exact Gauss-Bonnet ΣΦ_i=4π.
Conclusion: Validated degenerate triangles act as UV regulator, eliminating singularities while preserving continuum limit.
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Kalimuthu S. Validation of type I degenerated spherical triangles. Ann Math Phys. 2026.
Kalimuthu S. Validation of type II degenerated spherical triangles and tri-rectangular case. Ann Math Phys. 2026.
Girard A. Invention nouvelle en l’algèbre. 1629.
Regge T. General relativity without coordinates. Nuovo Cim. 1961;19:558-571.
Rovelli C, Smolin L. Discreteness of area and volume in quantum gravity. Nucl Phys B. 1995.
Ashtekar A, et al. Quantum nature of big bang. Phys Rev Lett. 2006.
Mayaan M. Mayamata and Pranava Veda (Vaastu Shastra).
Vethathiri Maharishi. Unified force. Vethathiri Publications; 1990.
Gauss CF. Theorema egregium. 1828.
Einstein A. Die Feldgleichungen der Gravitation. 1915.